Power Series Expansion for Real Area Hyperbolic Secant

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Theorem

The (real) area hyperbolic secant function has a Taylor series expansion:

\(\ds \arsech x\) \(=\) \(\ds \ln \frac 2 x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}! x^{2 n} } {2^{2 n} \paren {n!}^2 \paren {2 n} } }\)
\(\ds \) \(=\) \(\ds \ln \frac 2 x - \paren {\dfrac 1 2 \dfrac {x^2} 2 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^4} 4 + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^6} 6 + \cdots}\)

for $0 < x \le 1$.


Proof

From Power Series Expansion for Real Area Hyperbolic Cosine:

\(\ds \arcosh x\) \(=\) \(\ds \ln 2 x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } }\)
\(\ds \) \(=\) \(\ds \ln 2 x - \paren {\dfrac 1 2 \dfrac 1 {2 x^2} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {4 x^4} + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {6 x^6} + \cdots}\)

for $x \ge 1$.

From Real Area Hyperbolic Cosine of Reciprocal equals Real Area Hyperbolic Secant:

$\map \arcosh {\dfrac 1 x} = \arsech x$


So:

\(\ds \arsech x\) \(=\) \(\ds \ln \frac 2 x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}! x^{2 n} } {2^{2 n} \paren {n!}^2 \paren {2 n} } }\)
\(\ds \) \(=\) \(\ds \ln \frac 2 x - \paren {\dfrac 1 2 \dfrac {x^2} 2 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^4} 4 + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^6} 6 + \cdots}\)

For $1 \le \dfrac 1 x$ we have that $0 < x \le 1$.

Hence the result.

$\blacksquare$


Also see